Rough Sets Tutorial

Contents

Introduction Basic Concepts of Rough Sets Concluding Remarks

(Summary, Advanced Topics, References and Further Readings).

This is an abridged version of a ppt of 208 slides!!

Introduction

Rough set theory was developed by Zdzislaw Pawlak in the early 1980’s.

Representative Publications:– Z. Pawlak, “Rough Sets”, International Journal

of Computer and Information Sciences, Vol.11, 341-356 (1982).

– Z. Pawlak, Rough Sets - Theoretical Aspect of Reasoning about Data, Kluwer Academic Pubilishers (1991).

Introduction (2)

The main goal of the rough set analysis is induction of approximations of concepts.

Rough sets constitutes a sound basis for KDD. It offers mathematical tools to discover patterns hidden in data.

It can be used for feature selection, feature extraction, data reduction, decision rule generation, and pattern extraction (templates, association rules) etc.

identifies partial or total dependencies in data, eliminates redundant data, gives approach to null values, missing data, dynamic data and others.

Introduction (3)

Fuzzy Sets

Introduction (4)

Rough Sets– In the rough set theory, membership is not the

primary concept.– Rough sets represent a different mathematical

approach to vagueness and uncertainty.

Introduction (5) Rough Sets

– The rough set methodology is based on the premise that lowering the degree of precision in the data makes the data pattern more visible.

– Consider a simple example. Two acids with pKs of

respectively pK 4.12 and 4.53 will, in many contexts,

be perceived as so equally weak, that they are

indiscernible with respect to this attribute.

– They are part of a rough set ‘weak acids’ as compared to ‘strong’ or ‘medium’ or whatever other category, relevant to the context of this classification.

Basic Concepts of Rough Sets

Information/Decision Systems (Tables) Indiscernibility Set Approximation Reducts and Core Rough Membership Dependency of Attributes

Information Systems/Tables

IS is a pair (U, A) U is a non-empty

finite set of objects. A is a non-empty finite

set of attributes such that for every

is called the value set of a.

aVUa :

.AaaV

Age LEMS

x1 16-30 50x2 16-30 0x3 31-45 1-25x4 31-45 1-25

x5 46-60 26-49x6 16-30 26-49

x7 46-60 26-49

Decision Systems/Tables

DS: is the decision

attribute (instead of one we can consider more decision attributes).

The elements of A are called the condition attributes.

Age LEMS Walk

x1 16-30 50 yesx2 16-30 0 no x3 31-45 1-25 nox4 31-45 1-25 yes

x5 46-60 26-49 nox6 16-30 26-49 yes

x7 46-60 26-49 no

}){,( dAUT Ad

Issues in the Decision Table

The same or indiscernible objects may be represented several times.

Some of the attributes may be superfluous.

Indiscernibility

The equivalence relation A binary relation which is

reflexive (xRx for any object x) ,

symmetric (if xRy then yRx), and

transitive (if xRy and yRz then xRz). The equivalence class of an element

consists of all objects such that xRy.

XXR

Xx Xy

Rx][

Indiscernibility (2) Let IS = (U, A) be an information system, then

with any there is an associated equivalence relation:

where is called the B-indiscernibility relation.

If then objects x and x’ are indiscernible from each other by attributes from B.

The equivalence classes of the B-indiscernibility relation are denoted by

AB

)}'()(,|)',{()( 2 xaxaBaUxxBIND IS

)(BINDIS

),()',( BINDxx IS

.][ Bx

An Example of Indiscernibility The non-empty subsets of

the condition attributes are {Age}, {LEMS}, and {Age, LEMS}.

IND({Age}) = {{x1,x2,x6}, {x3,x4}, {x5,x7}}

IND({LEMS}) = {{x1}, {x2}, {x3,x4}, {x5,x6,x7}}

IND({Age,LEMS}) = {{x1}, {x2}, {x3,x4}, {x5,x7}, {x6}}.

Age LEMS Walk

x1 16-30 50 yes x2 16-30 0 no x3 31-45 1-25 nox4 31-45 1-25 yes

x5 46-60 26-49 nox6 16-30 26-49 yes

x7 46-60 26-49 no

Observations

An equivalence relation induces a partitioning of the universe.

The partitions can be used to build new subsets of the universe.

Subsets that are most often of interest have the same value of the decision attribute.

It may happen, however, that a concept such as “Walk” cannot be defined in a crisp manner.

Set Approximation

Let T = (U, A) and let and We can approximate X using only the information contained in B by constructing the B-lower and B-upper approximations of X, denoted and respectively, where

AB .UX

XB XB

},][|{ XxxXB B

}.][|{ XxxXB B

Set Approximation (2)

B-boundary region of X,

consists of those objects that we cannot decisively classify into X in B.

B-outside region of X,

consists of those objects that can be with certainty classified as not belonging to X.

A set is said to be rough if its boundary region is non-empty, otherwise the set is crisp.

,)( XBXBXBNB

,XBU

An Example of Set Approximation Let W = {x | Walk(x) =

yes}.

The decision class, Walk, is rough since the boundary region is not empty.

Age LEMS Walk

x1 16-30 50 yes x2 16-30 0 no x3 31-45 1-25 nox4 31-45 1-25 yes

x5 46-60 26-49 nox6 16-30 26-49 yes

x7 46-60 26-49 no

}.7,5,2{

},4,3{)(

},6,4,3,1{

},6,1{

xxxWAU

xxWBN

xxxxWA

xxWA

A

An Example of Set Approximation (2)

yes

yes/no

no

{{x1},{x6}}

{{x3,x4}}

{{x2}, {x5,x7}}

AW

WA

U

Set X

U/R

R : subset of attributes

XR

XXR

Lower & Upper Approximations

Lower & Upper Approximations (2)

}:/{ XYRUYXR

}:/{ XYRUYXR

Lower Approximation:

Upper Approximation:

Lower & Upper Approximations(3)

X1 = {u | Flu(u) = yes}

= {u2, u3, u6, u7}

RX1 = {u2, u3} = {u2, u3, u6, u7, u8, u5}

X2 = {u | Flu(u) = no}

= {u1, u4, u5, u8}

RX2 = {u1, u4} = {u1, u4, u5, u8, u7, u6}X1R

X2R

U Headache Temp. FluU1 Yes Normal NoU2 Yes High YesU3 Yes Very-high YesU4 No Normal NoU5 NNNooo HHHiiiggghhh NNNoooU6 No Very-high YesU7 NNNooo HHHiiiggghhh YYYeeesssU8 No Very-high No

The indiscernibility classes defined by R = {Headache, Temp.} are {u1}, {u2}, {u3}, {u4}, {u5, u7}, {u6, u8}.

Lower & Upper Approximations (4)

R = {Headache, Temp.}U/R = { {u1}, {u2}, {u3}, {u4}, {u5, u7}, {u6, u8}}

X1 = {u | Flu(u) = yes} = {u2,u3,u6,u7}X2 = {u | Flu(u) = no} = {u1,u4,u5,u8}

RX1 = {u2, u3}

= {u2, u3, u6, u7, u8, u5}

RX2 = {u1, u4}

= {u1, u4, u5, u8, u7, u6}

X1R

X2R

u1

u4u3

X1 X2

u5u7u2

u6 u8

Properties of Approximations

YX

YBXBYXB

YBXBYXB

UUBUB BB

XBXXB

)()()(

)()()(

)()(,)()(

)(

)()( YBXB )()( YBXB implies and

Properties of Approximations (2)

)())(())((

)())(())((

)()(

)()(

)()()(

)()()(

XBXBBXBB

XBXBBXBB

XBXB

XBXB

YBXBYXB

YBXBYXB

where -X denotes U - X.

Four Basic Classes of Rough Sets

X is roughly B-definable, iff and

X is internally B-undefinable, iff and

X is externally B-undefinable, iff and

X is totally B-undefinable, iff and

)(XB,)( UXB

)(XB,)( UXB

)(XB

,)( UXB )(XB

.)( UXB

Accuracy of Approximation

　 where |X| denotes the cardinality of

Obviously

If X is crisp with respect to B.

If X is rough with respect to B.

|)(|

|)(|)(

XB

XBXB

.X

.10 B,1)( XB,1)( XB

Issues in the Decision Table

The same or indiscernible objects may be represented several times.

Some of the attributes may be superfluous (redundant).

That is, their removal cannot worsen the classification.

Reducts

Keep only those attributes that preserve the indiscernibility relation and, consequently, set approximation.

There are usually several such subsets of attributes and those which are minimal are called reducts.

Dispensable & Indispensable Attributes

Let Attribute c is dispensable in T if , otherwise attribute c is indispensable in T.

.Cc

)()( }){( DPOSDPOS cCC

XCDPOSDUX

C /

)(

The C-positive region of D:

Independent

T = (U, C, D) is independent

　 if all are indispensable in T. Cc

Reduct & Core

The set of attributes is called a reduct of C, if T’ = (U, R, D) is independent and

The set of all the condition attributes

indispensable in T is denoted by CORE(C).

where RED(C) is the set of all reducts of C.

CR

).()( DPOSDPOS CR

)()( CREDCCORE

An Example of Reducts & Core

U Headache Musclepain

Temp. Flu

U1 Yes Yes Normal NoU2 Yes Yes High YesU3 Yes Yes Very-high YesU4 No Yes Normal NoU5 No No High NoU6 No Yes Very-high Yes

U Muscle pain

Temp. Flu

U1,U4 Yes Normal No U2 Yes High Yes U3,U6 Yes Very-high Yes U5 No High No

U

Headache Temp. Flu

U1 Yes Normal No U2 Yes High Yes U3 Yes Very-high Yes U4 No Normal No U5 No High No U6 No Very-high Yes

Reduct1 = {Muscle-pain,Temp.}

Reduct2 = {Headache, Temp.}

CORE = {Headache,Temp} 　 {MusclePain, Temp} = {Temp}

Discernibility Matrix (relative to positive region)

Let T = (U, C, D) be a decision table, with

By a discernibility matrix of T, denoted M(T), we will

mean matrix defined as:

for i, j = 1,2,…,n such that or belongs to the C-positive region of D.

is the set of all the condition attributes that classify objects ui and uj into different classes.

}.,...,,{ 21 nuuuU

nn

)]()([)}()(:{)]()([

jiji

ji

udud Dd if ucuc Ccudud Dd if ijm

iu ju

ijm

Discernibility Matrix (relative to positive region) (2)

The equation is similar but conjunction is taken over all non-empty entries of M(T) corresponding to the indices i, j such that

or belongs to the C-positive region of D. denotes that this case does not need to be

considered. Hence it is interpreted as logic truth. All disjuncts of minimal disjunctive form of this

function define the reducts of T (relative to the positive region).

iu juijm

Discernibility Function (relative to objects)

For any ,Uui

}},...,2,1{,:{)( njijmuf ijj

iT

where (1) is the disjunction of all variables a such that if (2) if (3) if

ijm,ijma .ijm

),( falsemij .ijm),(truetmij .ijm

Each logical product in the minimal disjunctive normal form (DNF) defines a reduct of instance .iu

Examples of Discernibility Matrix

No a b c d

u1 a0 b1 c1 yu2 a1 b1 c0 nu3 a0 b2 c1 nu4 a1 b1 c1 y

C = {a, b, c}D = {d}

In order to discern equivalence classes of the decision attribute d, to preserve conditions described by the discernibility matrix for this table

u1 u2 u3

u2

u3

u4

a,c

b

c a,b

Reduct = {b, c}

cb

bacbca

)()(

Examples of Discernibility Matrix (2)

u1 u2 u3 u4 u5 u6

u2

u3

u4

u5

u6

u7

b,c,d b,c

b b,d c,d

a,b,c,d a,b,c a,b,c,d

a,b,c,d a,b,c a,b,c,d

a,b,c,d a,b c,d c,d

Core = {b}

Reduct1 = {b,c}

Reduct2 = {b,d}

a b c d f u1 1 0 2 1 1 u2 1 0 2 0 1 u3 1 2 0 0 2 u4 1 2 2 1 0 u5 2 1 0 0 2 u6 2 1 1 0 2 u7 2 1 2 1 1

What Are Issues of Real World ?

Very large data sets Mixed types of data (continuous valued,

symbolic data) Uncertainty (noisy data) Incompleteness (missing, incomplete data) Data change

Use of background knowledge

A Rough Set Based KDD Process

Discretization based on RS and Boolean Reasoning (RSBR).

Attribute selection based RS with Heuristics (RSH).

Rule discovery by Generalization Distribution Table (GDT)-RS.

KDD: Knowledge Discovery and Datamining

Summary

Rough sets offers mathematical tools and constitutes a sound basis for KDD.

We introduced the basic concepts of (classical) rough set theory.

Advanced Topics(to deal with real world problems)

Recent extensions of rough set theory (rough mereology: approximate synthesis of objects) have developed new methods for decomposition of large datasets, data mining in distributed and multi-agent systems, and fusion of information granules to induce complex information granule approximation.

Advanced Topics (2)(to deal with real world problems)

Combining rough set theory with logic (including non-classical logic), ANN, GA, probabilistic and statistical reasoning, fuzzy set theory to construct a hybrid approach.

References and Further Readings Z. Pawlak, “Rough Sets”, International Journal of Computer

and Information Sciences, Vol.11, 341-356 (1982). Z. Pawlak, Rough Sets - Theoretical Aspect of Reasoning about

Data, Kluwer Academic Publishers (1991). L. Polkowski and A. Skowron (eds.) Rough Sets in Knowledge

Discovery, Vol.1 and Vol.2., Studies in Fuzziness and Soft Computing series, Physica-Verlag (1998).

L. Polkowski and A. Skowron (eds.) Rough Sets and Current Trends in Computing, LNAI 1424. Springer (1998).

T.Y. Lin and N. Cercone (eds.), Rough Sets and Data Mining, Kluwer Academic Publishers (1997).

K. Cios, W. Pedrycz, and R. Swiniarski, Data Mining Methods for Knowledge Discovery, Kluwer Academic Publishers (1998).

References and Further Readings

R. Slowinski, Intelligent Decision Support, Handbook of Applications and Advances of the Rough Sets Theory, Kluwer Academic Publishers (1992).

S.K. Pal and S. Skowron (eds.) Rough Fuzzy Hybridization: A New Trend in Decision-Making, Springer (1999).

E. Orlowska (ed.) Incomplete Information: Rough Set Analysis, Physica-Verlag (1997).

S. Tsumolto, et al. (eds.) Proceedings of the 4th International Workshop on Rough Sets, Fuzzy Sets, and Machine Discovery, The University of Tokyo (1996).

J. Komorowski and S. Tsumoto (eds.) Rough Set Data Analysis in Bio-medicine and Public Health, Physica-Verlag (to appear).

References and Further Readings W. Ziarko, “Discovery through Rough Set Theory”, Knowledge

Discovery: viewing wisdom from all perspectives, Communications of the ACM, Vol.42, No. 11 (1999).

W. Ziarko (ed.) Rough Sets, Fuzzy Sets, and Knowledge Discovery, Springer (1993).

J. Grzymala-Busse, Z. Pawlak, R. Slowinski, and W. Ziarko, “Rough Sets”, Communications of the ACM, Vol.38, No. 11 (1999).

Y.Y. Yao, “A Comparative Study of Fuzzy Sets and Rough Sets”, Vol.109, 21-47, Information Sciences (1998).

Y.Y. Yao, “Granular Computing: Basic Issues and Possible Solutions”, Proceedings of JCIS 2000, Invited Session on Granular Computing and Data Mining, Vol.1, 186-189 (2000).

References and Further Readings N. Zhong, A. Skowron, and S. Ohsuga (eds.), New Directions in

Rough Sets, Data Mining, and Granular-Soft Computing, LNAI 1711, Springer (1999).

A. Skowron and C. Rauszer, “The Discernibility Matrices and Functions in Information Systems”, in R. Slowinski (ed) Intelligent Decision Support, Handbook of Applications and Advances of the Rough Sets Theory, 331-362, Kluwer (1992).

A. Skowron and L. Polkowski, “Rough Mereological Foundations for Design, Analysis, Synthesis, and Control in Distributive Systems”, Information Sciences, Vol.104, No.1-2, 129-156, North-Holland (1998).

C. Liu and N. Zhong, “Rough Problem Settings for Inductive Logic Programming”, in N. Zhong, A. Skowron, and S. Ohsuga (eds.), New Directions in Rough Sets, Data Mining, and Granular-Soft Computing, LNAI 1711, 168-177, Springer (1999).

References and Further Readings

J.Z. Dong, N. Zhong, and S. Ohsuga, “Rule Discovery by Probabilistic Rough Induction”, Journal of Japanese Society for Artificial Intelligence, Vol.15, No.2, 276-286 (2000).

N. Zhong, J.Z. Dong, and S. Ohsuga, “GDT-RS: A Probabilistic Rough Induction System”, Bulletin of International Rough Set Society, Vol.3, No.4, 133-146 (1999).

N. Zhong, J.Z. Dong, and S. Ohsuga,“Using Rough Sets with Heuristics for Feature Selection”, Journal of Intelligent Information Systems (to appear).

N. Zhong, J.Z. Dong, and S. Ohsuga, “Soft Techniques for Rule Discovery in Data”, NEUROCOMPUTING, An International Journal, Special Issue on Rough-Neuro Computing (to appear).

References and Further Readings H.S. Nguyen and S.H. Nguyen, “Discretization Methods in Data

Mining”, in L. Polkowski and A. Skowron (eds.) Rough Sets in Knowledge Discovery, Vol.1, 451-482, Physica-Verlag (1998).

T.Y. Lin, (ed.) Journal of Intelligent Automation and Soft Computing, Vol.2, No. 2, Special Issue on Rough Sets (1996).

T.Y. Lin (ed.) International Journal of Approximate Reasoning, Vol.15, No. 4, Special Issue on Rough Sets (1996).

Z. Ziarko (ed.) Computational Intelligence, An International Journal, Vol.11, No. 2, Special Issue on Rough Sets (1995).

Z. Ziarko (ed.) Fundamenta Informaticae, An International Journal, Vol.27, No. 2-3, Special Issue on Rough Sets (1996).

References and Further Readings

A. Skowron et al. (eds.) NEUROCOMPUTING, An International Journal, Special Issue on Rough-Neuro Computing (to appear).

A. Skowron, N. Zhong, and N. Cercone (eds.) Computational Intelligence, An International Journal, Special Issue on Rough Sets, Data Mining, and Granular Computing (to appear).

J. Grzymala-Busse, R. Swiniarski, N. Zhong, and Z. Ziarko (eds.) International Journal of Applied Mathematics and Computer Science, Special Issue on Rough Sets and Its Applications (to appear).

Related Conference and Web Pages

RSCTC’2000 will be held in October 16-19, Banff, Canada

http://www.cs.uregina.ca/~yyao/RSCTC200/ International Rough Set Society

http://www.cs.uregina.ca/~yyao/irss/bulletin.html BISC/SIG-GrC

http://www.cs.uregina.ca/~yyao/GrC/

Thank You!

Rough Membership The rough membership function quantifies the

degree of relative overlap between the set X and the equivalence class to which x belongs.

The rough membership function can be interpreted as a frequency-based estimate of

where u is the equivalence class of IND(B).

Bx][

]1,0[: UBX |][|

|][|)(

B

BBX x

Xxx

),|( uXxP

Rough Membership (2) The formulae for the lower and upper approximations

can be generalized to some arbitrary level of precision by means of the rough membership function

Note: the lower and upper approximations as originally formulated are obtained as a special case with

]1,5.0(

}.1)(|{

})(|{

xxXB

xxXBBX

BX

.1

Dependency of Attributes

Discovering dependencies between attributes is an important issue in KDD.

Set of attribute D depends totally on a set of attributes C, denoted if all values of attributes from D are uniquely determined by values of attributes from C.

,DC

Dependency of Attributes (2)

Let D and C be subsets of A. We will say that D depends on C in a degree k

denoted by if

where called C-positive

region of D.

),10( k

,DC k

||

|)(|),(

U

DPOSDCk C

),()(/

XCDPOSDUX

C

Dependency of Attributes (3)

Obviously

If k = 1 we say that D depends totally on C. If k < 1 we say that D depends partially

(in a degree k) on C.

.||

|)(|),(

/

DUX U

XCDCk